The search for non-centrosymmetric superconductors that may exhibit unusual physical properties and unconventional superconductivity has yielded the synthesis of a non-centrosymmetric phosphide Mg$_2$Rh$_3$P with an Al$_2$Mo$_3$C-type structure. Although stoichiometric Mg$_2$Rh$_3$P does not exhibit superconductivity at temperatures above 2 K, we found that an Mg deficiency of approximately 5 at.% in the Mg$_2$Rh$_3$P induced superconductivity at 3.9 K. Physical properties such as the lattice parameter a = 0.70881 nm, Sommerfeld constant $gamma_n$ = 5.36 mJ mol$^{-1}$ K$^{-2}$, specific heat jump $Delta$C$_{el}$/$gamma_n$Tc = 0.72, electron-phonon coupling constant $lambda$$_{e-p}$ = 0.58, upper critical field H$_{c2}$(0) = 24.3 kOe, and pressure effect dTc/dP = -0.34 K/GPa were measured for the superconducting Mg$_{2-delta}$Rh$_3$P ($delta$ $sim$ 0.1). Band-structure calculations indicate that exotic fermions, which are not present in high-energy physics, exist in Mg$_2$Rh$_3$P. Since Mg, Rh, and P are the first elements used at each crystal site of Al$_2$Mo$_3$C-type compounds, the discovery of Mg$_2$Rh$_3$P may guide the search for new related materials.
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